I had this problem briefly after reinstalling my OS (windows 7 64bit). Xinput error every time I tried to open dolphin, and I did manage finding a solution to my particular problem with updating DirectX (DirectX End-User Runtime).

Note, this alone did not work as I got an error saying the application was unable to start as said in previous posts (same error too I believe). I then booted windows up into safe mode an installed it that way with no problems what so ever. No error with the directx app, no error for Xinput after updating, and most importantly didn't cause any more problems after doing this. Dolphin is currently running with no problems at all for me, nor did it start new problems.

If your thinking of simply using a Xinput1_3.dll in the right place, this did not work in my experience. I also do not know where the topic starter can find the dll as I fixed this some time ago.